of Microprocessors

PERCUSSIVE SOUND GENERATION
537
BOOM
ENVELOPE
WHITE
NOISE
OUTPUT
Fig. 15-4. Bass drum simulator
Nonlinear Vibrator Simulation
Many interesting percussive sounds fall into the Type 4 category. The
vibrating ruler mentioned earlier is one example, while a strongly plucked
rubber band is another. To these everyday examples can be added any
number of artificially contrived examples.
Such sounds involve one or more nonlinear vibrating members. One
characteristic of nonlinear vibrators is that the waveform and frequency of
vibration depend to some degree on the amplitude of the vibration. A linear
vibrator, on the other hand, is totally independent of amplitude. While
every natural (and electrical analog) vibrator is nonlinear to some extent, the
effect at normal amplitude levels is small enough to ignore.
The object here is to simulate the behavior, that is, plot the vibration
waveform, of an excited nonlinear vibrator given certain vital statistics about
its components and an expression or a table describing the nonlinearity. Figure
15-5 shows a standard spring-mass vibrator, just as it would appear in a
physics textbook. The position of the mass relative to its stable or neutral
position as a function of time is the variable of interest.
The classic first step in analyzing the vibrator is to note all of the forces
acting on the mass and then apply the conservation principle that requires
these forces to balance out, that is, sum to zero. There are three forces at
work (gravity will be ignored, since it is a static force that has no effect on the
vibration dynamics): the spring-restoring force, the force of friction, and the
force of inertia. The restoring force is normally proportional to the difference
between the present position and the neutral position and always pulls toward
the neutral position. The force-versus-position relation is what will be
made nonlinear later. The friction force can either be due to sliding friction,
in which case its magnitude is constant (as long as there is movement) or due
to viscous friction with magnitude proportional to velocity. Viscous friction
will be used, since it is better behaved and more "natural." In either case, the
direction is opposite to the direction of movement. The inertia force is
proportional to the mass of the vibrator and the acceleration (rate of speed